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We define $\\alpha=a/b, ~\\beta_n {=}b\\sqrt{2m U_n}/\\hbar$, we find $\\beta_n$ values for which critically the well has $n$-node half bound state at $E{=}0$. Consequently, this fixed well has $n$ number of bound states. Also we obtain a semi-classical expression ${\\cal G}(\\alpha,\\beta)$ such that the Fermi well has either $[\\cal G]$ or $[{\\cal G}]+1$ number of bound states. 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